A seven segment display is made up of seven individual LEDs (light emitting diodes) arranged in a figure-8 pattern.
Each segment is labeled A through G.
By turning on specific segments, you can display numbers and some letters.
"Common Cathode" means all the LED cathodes (negative terminals) are connected together and grounded (connected to 0V).
To light up a segment, you apply a HIGH voltage (positive) to the individual segment's anode (positive terminal).
Opposite: A "common anode" display has all the anodes tied together, and the segments are turned on by pulling their cathodes LOW.
This refers to the height of the digit displayed, not the size of the whole component.
0.56 inches ≈ 14.2 mm
It’s a fairly common size for general use — big enough to be easily seen, small enough for compact circuits.